Local Uniqueness for Nash Solutions of Multiparameter Singularly Perturbed Systems

Abstract

In this brief, linear quadratic infinite-horizon Nash games for general multiparameter singularly perturbed systems are studied. The local uniqueness and the asymptotic structure of the solutions to the cross-coupled multiparameter algebraic Riccati equation (CMARE) are newly established. Utilizing the asymptotic structure of the solutions to the CMARE, the parameter-independent Nash strategy is established. A numerical example is given to demonstrate the efficiency and feasibility of the proposed analysis.Index Terms-Cross-coupled algebraic Riccati equation (CARE), general multiparameter singularly perturbed systems (GMSPS), local uniqueness, Nash games, parameter-independent Nash strategy.

Stabilizing composite control for systems modeled by singularly perturbed Itô differential equations with two small time constants

“…Moreover, applying Proposition 2 (i), we deduce that (X 0 ,X 0f ,X f ) is the unique solution of the system (19). Taking into account that (19) is obtained from (16) for ν =ν, we deduce via implicit function theorem [9] that there exist ν 1 and the analytic function ν → (X 0 (ν), X 0f (ν), X f (ν)) defined for ν in the ball ||ν −ν|| < ν 1 which verify the system (10) and…”

“…With these notations, the system of type (16) corresponding to the pair F = F s F f can be written in the compact form:…”

“…Recent advance in theory of the stochastic approach has allowed a revisiting of the control problems for the MSPS [16], [17]. These literatures, however, the special structure for the fast subsystems are imposed.…”

Numerical solutions of soft constrained nash games for multiparameter singularly perturbed systems

“…Under Assumptions 1 and 2, using the similar technique in [9], [10], [11], the following zeroth-order equations of the GCSMARE (11) are given as ||ν||…”

“…Recently, the numerical approaches for solving the cross-coupled multiparameter algebraic Riccati equation (CMARE) related to the Nash games have been developed [9], [10]. However, for the previous results [1], [2], [9], [10], [11], the dynamic model of the environment in which the players act is assumed to be exact. That is, the disturbance input is not considered.…”